A035939 Number of partitions of n into parts not of the form 7k, 7k+3 or 7k-3. Also number of partitions such that the differences between parts at distance 2 are greater than 1.

1, 1, 2, 2, 3, 4, 6, 7, 10, 12, 16, 19, 25, 30, 38, 46, 57, 68, 84, 99, 121, 143, 172, 202, 242, 283, 336, 392, 462, 537, 630, 729, 851, 982, 1140, 1312, 1518, 1741, 2006, 2295, 2635, 3007, 3442, 3917, 4470, 5077, 5776, 6545, 7429, 8399, 9510, 10731

Offset: 0

Author

Olivier Gérard

Keywords

• nonn
• easy

Comments

Case k=3, i=3 of Gordon Theorem.

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Examples

			G.f. = 1 + x + 2*x^2 + 2*x^3 + 3*x^4 + 4*x^5 + 6*x^6 + 7*x^7 + 10*x^8 + ...
G.f. = q^-1 + q^41 + 2*q^83 + 2*q^125 + 3*q^167 + 4*q^209 + 6*q^251 + ...
		

References

• G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976, p. 109.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Michael Somos, Introduction to Ramanujan theta functions
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Crossrefs

Cf. A035937, A035938.

Programs

Maple

# See A035937 for GordonsTheorem
A035939_list := n -> GordonsTheorem([1, 1, 0, 0, 1, 1, 0], n):
A035939_list(40); # Peter Luschny, Jan 22 2012

Mathematica

a[ n_] := SeriesCoefficient[ 1 / Product[ (1 - x^(7 k - 1)) (1 - x^(7 k - 2)) (1 - x^(7 k - 5)) (1 - x^(7 k - 6)), {k, Ceiling[n/7]}], {x, 0, n}]; (* Michael Somos, Dec 29 2014 *)
a[ n_] := SeriesCoefficient[ 1 / (QPochhammer[ x^1, x^7] QPochhammer[ x^2, x^7] QPochhammer[ x^5, x^7] QPochhammer[ x^6, x^7] ), {x, 0, n}]; (* Michael Somos, Dec 29 2014 *)

PARI

{a(n) = my(A); if( n<0, 0, polcoeff( 1 / prod( k=1, n, 1 -[0, 1, 1, 0, 0, 1, 1][k%7 + 1] * x^k, 1 + x * O(x^n)), n))}; /* Michael Somos, Dec 29 2014 */

Sage

# See A035937 for GordonsTheorem
def A035939_list(len) :  return GordonsTheorem([1, 1, 0, 0, 1, 1, 0], len)
A035939_list(40) # Peter Luschny, Jan 22 2012

Formula

Expansion of f(-x^3, -x^4) / f(-x, -x^2) in powers of x where f() is Ramanujan's two-variable theta function. - Michael Somos, Dec 29 2014

Euler transform of period 7 sequence [ 1, 1, 0, 0, 1, 1, 0, ...].- Michael Somos, Dec 29 2014

G.f.: 1 / (Product_{k>0} (1 - x^(7*k - 6)) * (1 - x^(7*k - 5)) * (1 - x^(7*k - 2)) * (1 - x^(7*k - 1))). - Michael Somos, Dec 29 2014

G.f.: (Product_{k>0} (1 + x^k)) * (Sum_{k>=0} x^(2*k^2) / (Product_{i=1..k} (1 - x^(2*i)) * (1 + x^(2*i-1)) * (1 + x^(2*i)))). - Michael Somos, Dec 31 2014

a(n) ~ 2^(1/4) * cos(Pi/14) * exp(2*Pi*sqrt(2*n/21)) / (3^(1/4) * 7^(3/4) * n^(3/4)). - Vaclav Kotesovec, Nov 13 2015

Extensions

Missing a(0)=1 prepended by Michael Somos, Dec 29 2014

Name simplified by George Beck, Aug 27 2023